CRYSTALLOGRAPHY, GEOMETRY AND PHYSICS IN HIGHER DIMENSIONS .11. A NEW GEOMETRICAL METHOD FOR SYSTEMATIC CONSTRUCTION OF THE N-DIMENSIONAL CRYSTAL FAMILIES - REDUCIBLE AND IRREDUCIBLE CRYSTAL FAMILIES

A geometrical method is described for counting and constructing all crystal families of Euclidean spaces of dimension n from the different well known crystal families of spaces E1, E2, E3 and the geometrically Z-irreducible families of each space. The definition of the geometrically Z-irreducible (gZ-irr.) or geometrically Z-reducible (gZ-red.) families is connected to the properties of the character table of the holohedry of these families. Indeed, a crystal family of space E(n) is said to be gZ-irr. if the n translation operators corresponding to a basis of a primitive Bravais cell belong to the same irreducible representation with integer entries of its holohedry. In the opposite case, the family is said to be gZ-red. This method enables a name to be given to the crystal families. This name is connected to the geometrical construction, except for the families considered as irreducible. As far as possible, it also recalls the name of the crystal families of spaces E1, E2 and E3. Moreover, the WPV (Weigel-Phan-Veysseyre) symbols of the holohedries can be defined thanks to the properties of the crystal cells. Finally, all the point-symmetry operations of these groups and subgroups can be listed.